Fixed point theory for Jaggi L-type mappings

A synthetic axiomatization of Map Theory

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

On the topological degree for A-compact mappings

Fixed points of [formula omitted]-valued maps, the fixed point property and the case of surfaces—A braid approach

We introduce a very general class of generalized non-expansive maps. This new class of maps properly includes the class of Suzuki non-expansive maps, Reich–Suzuki type non-expansive maps, and generalized α -non-expansive maps. We establish some basic properties and demiclosed principle for this class of maps. After this, we establish existence and convergence results for this class of maps in the context of uniformly convex Banach spaces and compare several well known iterative algorithms.

Interative approximation of fixed points for almost asymptotically nonexpansive type mappings in Banach spaces

One of the most famous results of topological fixed-point theory is the Lefschetz–Hopf fixed-point theorem. Many generalizations of this theorem to various classes of maps and spaces were obtained in the last decades. Compactness plays an essential role in these generalizations. We show here how the classical compactness conditions may be weakened to conditions which are necessary or almost necessary. Recall that a topological space has the fixed-point property if and only if any continuous map of the space into itself has a fixed point. Frequently the Lefschetz–Hopf theorem is used to prove the fixed-point property for compact spaces with an additional structure, e.g. polyhedra, quasi-complexes, ANR’s, semicomplexes and so on. The well-known Brouwer fixed-point theorem may be obtained in the same way.

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂn, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.

A class of biholomorphic mappings named quasi-convex mapping is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to Cn, the quasi-convex mapping is exactly the quasi-convex mapping of type A introduced by K.A. Roper and T. J. Suffridge.

In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighborhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as the main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]).
 Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved (comp. [11], [13]-15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1−n)-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighborhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.

In this paper we consider several classes of mappings related to the class of contraction mappings by introducing a convexity condition with respect to the iterates of the mappings. Several fixed point theorems are proved for such mappings. Further, in a similar way we consider a related class of mappings satisfying a convexity condition with respect to diameters of bounded sets. In the last part we consider classes of mappings on PM- spaces (probabilistic metric spaces of K. Menger) and some fixed point theorems are given for such classes.

We introduce the concept of ψ-firmly nonexpansive mapping, which includes a firmly nonexpansive mapping as a special case in a uniformly convex Banach space. It is shown that every bounded closed convex subset of a reflexive Banach space has the fixed point property for ψ-firmly nonexpansive mappings, an important subclass of nonexpansive mappings. Furthermore, Picard iteration of this class of mappings weakly converges to a fixed point. MSC:47H06, 47J05, 47J25, 47H10, 47H17.

Introduction Let K be a compact convex subset of a real topological vector space E, which we shall always assume to be separated (i.e. Hausdorff). We consider multi-valued mappings T of K into E, i.e. mappings (in the usual sense) of K into 2 e, the space of subsets of E, where for each x in K, T(x) is a non-empty closed convex subset of E. By a fixed point of such a mapping, we mean a point u of K such that u e T(u). The earliest extension of the topological theory of fixed points of continuous mappings to the case of multi-valued mappings was made by von Neumann [27] in the connection with the proof of the fundamental theorem of game theory. The extension of the Brouwer fixed point theorem to an upper semi-continuous multivalued mapping T of a n-disk into itself was carried through by Kakutani [23] and corresponding extensions of the Schauder fixed point theorem in Banach spaces were given independently by Bohnenblust-Karlin [33 and Glicksberg [193. The corresponding extension of TychonolTs theorem for locally convex topological vector spaces was proved by Ky Fan [12], who in a group of subsequent papers ([13, 14, 15, 16, 17]) refined and extended this result and considered a variety of applications. Asymptotic fixed point theorems for multi-valued mappings in Banach spaces were established in Browder [4], and parts of the Leray-Schauder theory in Banach spaces were extended to multi-valued mappings by Granas [20, 21]. It is our object in the present paper to present a new general treatment of the fixed point theory of multi-valued mappings in topological vector spaces which has the dual virtues of obtaining new and stronger results on the one hand and drastically simplifying the proofs of known results on the other. The starting point of our investigation of this theory lies in recent results of the writer in connection with the study of monotone operators and non-linear variational inequalities [6, 7, 8]. In this direction, one considers mappings S of a compact convex set K into E*, the dual space of E, rather than E itself. Instead of trying to find fixed points of a mapping T of K into E, one looks for points u in K for which S(u)= 0, or more generally, for which

In this paper, a new class of mapping that unifies various classes of mappings associated with the class of asymptotically nonexpansive mappings is introduced. In addition, an iterative technique for approximation of fixed points of this class of mappings is introduced and studied in the setting of uniformly convex real Banach space. Moreover, Demiclosedness principle for the class of mapping under study is proved; in addition, weak and strong convergence theorems are obtained. The theorems obtained augment, generalize, improve and unify several results that are recently announced. The method of proof used is of independent interest.

A new class of fuzzy contractive mappings and fixed point theorems